Good morning, everybody, and welcome to the next lecture
on time series equal matrix.
May I ask you if there are any questions?
Then please raise your hand.
I do not see any questions.
So let me repeat what we did last week
because this part of the lecture is really important
for anybody who wants to work in time series
and with time series.
So work on an applied level.
You really need to know what I did last week
and what I will do in today's lecture.
What I taught you was basically by means of examples
with just a little bit of theoretical background
and that two time series which have a unit root
and for simplicity we just took time series
which were random walks, but the same thing
holds for general unit root processes.
That two time series which have a unit root
but which have nothing to do with each other
which have been generated completely independently
of each other would if one of the time series,
one of the unit root processes is regressed
on the other unit root process
with a great likelihood indicate
that there is a relationship between the two random walks.
So would give with standard methods of inference
a completely wrong picture of what truly is the case.
In particular, we would find that the regression coefficient
of the regressor, the random walk of the unit root process
which serves as the regressor
that its coefficient would have a very high T statistic.
In our example here you see the T statistic is 42.
So actually one would believe this beyond any doubt
that random walk one and random walk two
are related to each other probably with some type of causality.
And the R square would be very high
if we take two random walks with drift
or two unit root processes.
Now we analyze this a little bit
and I told you that basically what happens
is that we have trending data in this example
and macroeconomists very often work with trending data.
Actually I warned you against deep trending data
because here again the unit root processes cause great trouble
and if we have a process
which is truly a unit root process and trending
then deep trending might induce spurious cycles
in a series which actually doesn't have any cycles.
So macroeconomists when they see trending data
are warned against just deep trending the data
at least not the trending the data
by just regressing them on a linear trend
as people have done before.
But if we take two trending series
and redress them one on the other
then we may or we will see a significant test statistic
even though the two series
may not have anything to do with each other
just because well there are two trends
and one trend seems to explain the other
although trends are perhaps of a completely different origin.
So it's just that they trend
which causes a significant regression coefficient
and not that the share is in an economic sense
in a real world sense that the trend
and then in a real world sense is the same.
It is just the case that these trends
both feature in the time series
and so the regression result seems to be significant.
A famous example of this type of spurious regression
was given years ago by people who were looking
at the number of births in Sweden
and the number of storks birds in Sweden.
The white storks which people thought
or which people told the children would bring the babies.
So obviously storks have nothing to do with human births
but to sort of illustrate what the problem is
a regression of births in Sweden
on the number of storks in Sweden
produces a highly significant estimate.
Simply because over years and perhaps decades
births were decreasing in Sweden
so numbers were going down
the number of numbers of storks was also going down
for of course completely different reasons
but trend explains trend and so a regression of one variable
on the other produced a significant coefficient
which is an example which should warn you
of over interpreting significant coefficients.
You may easily get them between two time series
which have nothing to do with each other
and that is the same thing here where we regressed last week
two random walks on each other
and found that the test statistic is significant
for the second random walk
so in this case it was random walk number one
as a regressor in a regression of random walk number two on number one.
Now this seems easy because you may think okay
trend explains trend but the problem is even more intricate
because when we take two random walks without drift
two random walks which do not have a drift
and regress them on each other
and this was this example which I gave you here
then we see graphically quite clearly
that there's little relationship between random walk one
and random walk two.
Nevertheless when we regress them on each other
we are still quite likely to observe a significant t value
for this regression coefficient here
which is the coefficient of the regressor random walk one
in your regression of random walk two on the constant and random walk one.
You see that the statistic in my example was 3.2
so at conventional significance levels
we would say this is highly significant
I will teach you today that this is not at all significant
at the one percent level actually that this t value here is useless
we cannot interpret it
so we are trying today to understand the spurious regressions
which I showed you last week
and in order to understand them I just repeat what we have been doing
so we see here in this particular example
where we have two random walks without drift data which are not threatening
but still we have the problem that regression results may be highly misleading
if we are not careful in looking at the unit root problem
and this what I show you here for random walks
generalizes to any process which has a unit root
so if you just take two processes with unit roots
and regress them on each other
then the coefficient of the regressor can be very misleading
and indicate some relationship between the dependent variable
and the independent variable which just does not exist in reality
and we had a little bit of discussion about why this is so
um the reason basically is that the regression which we estimated
regress one random walk you denoted yt
on the second random walk you denoted xt and on a constant
that this regression which we estimate does not admit the true model
because the true model for the true to random walks at least
was given by equations 15 and 16
and neither equation 15 nor equation 16 is nested in equation 17
so actually the data have no possibility of showing us what the true model is
so the data choose some type of parameterization
which minimizes the estimated residuals
but which is nonsensical because it cannot be the true parameterization
it is necessarily a wrong parameterization
and given that the data are forced to assume some type of wrong parameterization
it may happen and not with a small likelihood
that the parameters which the data choose in order to minimize the sum of the squared
residuals that these parameters actually appear to be significant
we will discuss the true distributions today and we will see that actually if we apply correct
correct inference then the significance vanishes and basically just in perhaps five percent of the
cases we find a significant t statistic but we will have to use different distributions we cannot
use the t distribution anymore as you will see so that's the important lesson from equation 17
this equation here is misspecified regardless of whether we assume that beta one is zero
or beta one is non-zero regardless of what the data decide to suggest to us
beta one is equal to zero or beta one is not equal to zero both parameterizations would be
wrong they would not give us the correct parameterization or the correct equation
which is given in equation 16 here
all right so we went through this and i gave you some theoretical background to this
and then i stopped with some exercises now in particular i asked you to run a simulation study
on your computer and if you haven't done it yet or if you think well this is just an exercise
i would like to indicate that this is really an exercise where i can learn a lot
i mean even if you perhaps decided to to ignore all the exercises which i have given you
so far and in particular the computer exercises i really would suggest to you that you do this
exercise and the exercise on the next page because you will get a very good feeling of what the
problem with unit root processes is or what problems are because there are many of them
with unit root processes if you just do these exercises which are actually if you can do a
little bit of programming quite easy to do so since i will start with these exercises since
i have done similar simulation studies whose results i will now present to you let me just
read again what this exercise is about and what the exercise on the next slide is about
what i ask you to do is to do experimentally what i showed you by means of just one example
already here in the lecture first you generate independent random walks and you call them x and
you call them y so they use different random numbers which you generate in your computer
and then you have one random walk x and one random walk y and they have nothing to do
with each other good for instance for 100 observations but you can take as many observations
as you want to and do not provide a drift term so exclude the possibility that it is just
the trending behavior of the data which leads to significant statistics but do the pure random
walks two pure random walks which have nothing to do with each other and each of which has 100
observations and then compute the correlation rho between x and y and i'll do this for say
a great number of times like a thousand times each time you start out with generating new
random walks x and y which have nothing to do with each other i write here and i i notice this
description is not completely correct what i write there i write the asymptotic standard
error for the null hypothesis that rho is equal to zero is one over the square root of t
this would be true if we had two stationary series it is not true actually for two random walks so
what i should write here which what i will correct is the asymptotic standard error in the case that
x and y are stationary processes for this null hypothesis is one over square root of t which is
something which we derived in the earlier part of the lecture so one over the square root of t is
of course 0.1 right with 100 observations so you may now check after you computed the correlation
rho between x and y for thousand pairs of completely independent random walks
you may now compare these correlations with the asymptotic standard error of 0.1
so you will check how often does the correlation which you have computed actually exceed
0.2 so how often does it exceed two standard errors and you should find that it exceeds this
two standard error margin the two standard error margin which is derived from stationary theory
that exceeds this interval a great many times and much more than five percent
so plot the distribution of the rows and then in par p you may repeat the exercise
which is just changing one parameter in your program and then running it again
by generating not say a thousand pairs of random walks but generate ten thousand or even a hundred
thousand oh no sorry repeat the exercise with ten thousand observations per random
walk or with hundred thousand with sorry a thousand or with ten thousand or even with
hundred thousand observations per random walk to check if this problem that we find
spuriously high rows vanishes over time
okay same exercise run similar simulation study basically do the same thing again generate
two independent random walks with hundred observations each and then drift and now just
regress the two on each other so regress y on x and on a constant of course and compute the t
statistic for the coefficient of x basically you estimate the same thing as the row because as you
know in the bivariate regression the regression coefficient for the second regressor which is not
the constant is very closely related to the correlation between the two variables and then
you'll see that the distribution is actually quite similar now compute the t statistic for x and do
this for say a thousand times and then check how often you find a significant
since the two random walks x and y are independent you should actually expect that you find a
significant t statistic at the same five percent level only five percent of the cases but you will
see and i show the result to you in a minute but try to reproduce it on your own you will see that
you find significant t statistics much more often than five percent of the time
so basically this has the t distribution is not the correct distribution we cannot take
the critical values of the t statistic from the t distribution well then check again whether this
problem vanishes asymptotically if you take a thousand observations
and then perhaps do the same thing again and allow for drifts and you may take completely
different drifts i suggest here 0.2 for x and negative 0.25 for y but you can choose any number
and see how the problem changes when there are drift terms in the data generating process
of the random walks and therefore trends in the data which we observe
now as i said i have of course done this exercise and i've done many many times
my analysis used random walks x and y which are completely independent of each other and each of
which has 100 observations is sort of the number of observation which we rather typically have
in macroeconomics so this order of magnitude is quite typical for for the time spans that we
observe and then i let my regression or my simulations run through hundred thousand runs
so all over my computer generated 200 000 random walks right each of them were completely
independent of all the others so a hundred thousand times i regressed y on x knowing that
x does not have anything to do with y and then i computed how often my t test for the constant
beta zero and for the coefficient of the regressor of the random walk on x was significant
so here are the t test based rejections of the null hypothesis that beta not is equal to zero
and you see that in 84 percent of my hundred thousand runs so in 84 000 runs i found a
significant constant which perhaps is not yet too interesting and not yet too important but
it correlates with the fact that the t test based rejections of h zero of the null hypothesis
of beta one is equal to zero was 76 percent so in three quarters of the cases i found that x
is significant in the regression of y on x even though x has nothing to do with x in 76 percent of
the cases i had a t statistic which was greater than two this is what i mean by t test based
rejections of h not so of the null hypothesis right in 76 percent of the cases standard inference
which i have taught you for stationary time series which assumes that we have a rejection
at the five percent level when the t statistic is approximately two or greater than two the standard
inference would yield in this setting here and 76 percent of the cases we find that x is significant
in the regression of y on x or when we find something which will become more important later
because we can base a first test on it we find that the derm watson statistic is significant
all the time and 100 percent of my runs the derm watson statistic was smaller than 1.5
so it indicated positive first order autocorrelation in this regression
and found that the r squared was substantial for instance greater than 0.5 in 60 percent of the
cases we'll see this uh graphically in a minute but basically this tells us a researcher which
naively regresses y on x not knowing that x and y have been generated completely independently of
each other and have nothing to do with each other such a researcher might look at his regression
and might say hey i found a significant coefficient for beta 1 and i found that my regression explains
me more than 50 percent of the variance or more than 40 or more than 60 percent of the variance so
0.5 here is not some type of critical value just some indication that we may find a rather high
r squared and we may also find that the p value of the test statistic is lower than five percent
indicating to us that at least one of those two coefficients here is non-zero
this again happened in 76 percent of the cases any questions so far
raise your hand please if there are any questions
i don't see any indication so let's move on this here is the result of my simulations
where i just show you what coefficient beta 0 i have been estimating so this is the estimated
constant not yet the p statistic we come to this this statistic in a minute but just the estimated
constant what we see here first is nice that um the distribution is symmetric and it is symmetric
about zero where you may say well this is perhaps the most reasonable thing to have
um a constant of zero because as we know y and x have been generated independently of each other
there is no constant in play in the data generating process so it is centered on zero
and then of course there are estimates which are not exactly uh zero and that seems fine and you
may say where's the problem the problem is in the fact that some of the betas have been estimated
rather large so far off zero you see here is negative 10 here is plus 10 there are quite a
few values between zero and 10 which have been estimated and already this may be an indication
that people may think well we we have some type of imperfect constant here the true problem of
course is visible only when you look at the p statistic for the hypothesis that beta zero is
equal to zero so if we test the significance of beta zero in this case we see that the t statistic
has this nice and symmetric distribution or it looks nice at first sight but remember a t
statistic is typically significant or in standard inference and i have to emphasize here in standard
inference a t statistic is considered significant if it is greater than two an absolute value
so less than negative two or more than plus two and you see here two is somewhere here where my
my cursor hand currently is because here is already negative 20 and plus 20 so if just this range
here where my hand now moves is the range where the t statistic is less than two an absolute value
then we see we have a great number of estimates for the constant where the t statistic
is greater than two an absolute value so where the t statistic seems to indicate
significance of the constant
now the same thing holds true for the estimated beta again we see when we just look at the
estimated coefficient that is centered on zero the distribution is approximately symmetric between
values which are negative estimates which are negative and estimates which are positive
even though well some of those estimates seem to be quite far off zero we could still think okay
things like that happen but look at the t statistics for this coefficient and then you see
the same thing as for the constant that the t statistic is between negative two and two
this is the small area down here where my hand currently moves all the other t statistics would
be significant at conventional or significance levels so you see there are t statistics which
are in the area of magnitude of negative 20 or plus 20 so really there are many many t statistics
which seem to be significant and this is of course then misleading
now then here look at the derm Watson statistic for my hundred thousand regressions and how they
are distributed recall the derm Watson is a statistic which should be two if there is no auto
correlation in the resilience of the regression no auto correlation of first or valid same
and quite approximately there is some type of
region where we would find a deviation from the value of who is not significant which is
approximately between 1.6 and 2.4 but values of the derm Watson statistic which are
smaller than 1.5 say certainly indicate that the residuals are not white noise but are positively
auto coordinate now you see the derm Watson statistic here is far off its value too which
is somewhere somewhere right here most of the derm Watson statistics which my regressions have
produced are actually in the range between zero and 0.5 so far off really far off derm Watson
statistic of two which we would usually aim for and here's the distribution of the r squared
you see in many cases the r squared is actually zero and that's fine that is what we would expect
when we regress one variable on the second variable which has nothing to do with the first
variable but in many other cases actually the r squared is much greater or greater than
than zero the distribution is decays here in some way but you see
we can get many many r squares which are say greater than 0.4 or 0.5
so that again can be greatly misleading that the r squared is possibly great
even though the two time series x and y have nothing to do with each other
and then finally here's the p value of the f test which also has some type of distribution but you
see in many many cases the p value of the f test is just zero so indicates that either beta 0 or
beta 1 are significant are necessary in this nonsense or regression of two variables on each
other where variable x doesn't explain cannot explain variable y okay let me just open the
window because it's getting hot here
now the question is what is the reason for this peculiar behavior of our regression results
because what i now showed you in this simulation study is very typical and you can easily reproduce
it on your own computer the first systematic investigation of spurious regressions is credited
to Granger the later Nobel laureate and Newbold from 1974 but their paper at the time didn't
completely explain the issue and many people weren't really aware of the severity of the
problem which Granger and Newbold had already analyzed the breakthrough in understanding
spurious regressions occurred much later namely by Peter Phillips famous paper of 1986
understanding spurious regressions and this paper is highly technical actually its mathematics is
much beyond the level of this lecture here but i will try to give you at least some sense of what
Peter Phillips derived for this i just have to remind you that we had different concepts of
distribution of convergence of convergence of random variables and one of these concepts
was the convergence in distribution our definition was a series of random variables xn converges in
distribution to a random variable and its distribution if the limit of the distributions
of the expense is f of x and f of x is of course the distribution of the random variable x
as i write here fn and f are the cumulative distribution functions of xn and x and then
we write in shorthand notation that xn converges in distribution to random variable x
convergence in distribution is quite a often used a frequently used convergence concept in
econometrics we have made use of it for instance in the central limit theorem um even though in
many other cases we have used the plim convergence i also told you um already know it from from
basic econometrics the plim convergence is actually somewhat stronger than convergence
in distribution so plim convergence implies convergence in distribution but the reverse
is not true so uh convergence distribution does not imply plim convergence
now what peter phillips did in his 1986 paper was that he uses some kind of non-standard asymptotic
distribution theory this as i say um is deeply related to brownian motions and vina processes
concepts which i cannot cover in this vector here and which are fairly advanced um so i just
give you a sketch here of what peter phillips does in order to understand or at least follow
phillips results let us denote by script d v index i some type of non-degenerate non-standard
distribution so you know some of the standard distributions we have particular symbols for
we usually usually use a script n for the normal distribution or we use chi for chi squared
distribution or we use f for the f distribution and t for the t distribution and here d i shall
just mean distribution i some type of distribution and the only thing i require is uh or the only
thing i associate with uh d i is that this is a non-degenerate distribution meaning the
distribution is truly a distribution which spreads out somehow a degenerate distribution
would be a distribution which finally converges to just a single point so this we want to rule out
our script d's here denote non-degenerate distributions and there should be non-standard
distribution that is the d's i will show you different types of d's the d's are not the normal
distribution not the t distribution not the f distribution not the chi squared distribution so
nothing which you are familiar with but completely different distributions and i will show them to you
a little later all right phillips first shows that the estimate of beta one so beta one hat
estimated by ols in a regression of one unit root process on a second unit root process
that this estimate converges in distribution against the non-standard non-degenerate
distribution and it gives the exact formula how this distribution is to be described mathematically
which i spare you here i just tell you there is one non-standard non-degenerate distribution
called the estimate beta one half i want how the distribution looks like you'll see
later but i don't give the mathematical expression uh for for this but i have by
a convention already said it is a non-degenerate distribution so dust spread out even asymptotic
what does this mean this means that when we have a spurious regression then beta one hat
does neither converge to a constant nor does it diverge recall in standard ols analysis
we can show that beta one hat has a plim so it converges against a constant but here in
constant but here in unit root econometrics when we have a regression between two unit root processes
then the coefficient which we estimate does not converge to constant but it converges
against the distribution does also not explode the coefficient but it converges against the
well-defined distribution and first derive what distribution this is
now what we also know from standard theory is that in the stationary case we would have
the property t that the square root of t times beta one hat converges in distributions against
the standard normal distribution so we have called this the property that beta one hat converges in
converges at a speed of one over the square root of t to the standard normal distribution
in unit root econometrics however things are almost the contrary in the spurious regression
case which we are currently studying the t statistic for beta one hat diverges at speed
square root of t so when i take the t statistic for beta one hat we call here
we have beta one hat which converges in distribution against a normal a non-standard
distribution and if i pull it apart with beta one hat by square root of t then it converges
against a normal distribution so this was the first set of result which i showed you from
peter village paper now in the case of the t statistic oh sorry where am i yeah in case of
the t statistic we see that t beta one hat the t statistic for beta one hat would explode
asymptotically would become larger and larger and only if i divide it by the square root of t
only then the t statistic converges against some distribution which here i call
script d2 because it's a different distribution from script d1 against which beta one hat converges
so i need to divide t beta one hat by square root of t in order to make sure that the t statistic
converges against a distribution a well-defined distribution rather than exploding
it says that if i do not have this factor t to the negative 0.5 if i had not had this
factor here for rescaling the t statistic then the t statistic would explode or would spread
out more and more which means that values greater than two an absolute value would be easily
observed so this means that there are no asymptotic particular values for t beta one hat
the same thing peter phillips actually shows for the estimate of the constant
so the estimate of the constant also needs to be divided by the square root of t
and only then does it converge to some type of non-degenerate non-standard distribution
the third type of distribution of this class of non-standard non-degenerate distributions
it's different from the distribution against which the beta one hat converged but again we see
that the estimate of the beta zero hat would actually diverge would actually explode if
we did not divide by the square root of t
this means that the estimate of the constant term diverges and it diverges at the speed square root
of t so the variance of beta zero hat so the variance of the estimated constant increases with
t and the same thing is true for the likelihood of extreme estimates beta zero hat
what happens to the t statistic of the beta zero hats this is what we have seen
here right t statistic of the beta zero hats
the t statistics of the beta zero hats diverges again and only if you divide it by square root of
t only then the t statistic of beta zero hat converges in distribution against
script d4 so another distribution which we do not know which is non-degenerate and non-standard
so again no asymptotic critical values for t test on beta zero hat exist
the r squared which i showed you which had this
first kind of funny looking shape here this distribution of the r squared
the r squared converges in distribution against again the non-standard distribution
which i denote here script d5 which is the fifth set distribution and different from all the
preceding ones and it converges against this distribution which has considerable likelihood
considerable probability mass so considerable support for positive values even asymptotically
so we may easily think that a regression indicates by means of its r squared some type of
explanatory power even though this is not true because we know now that they are squared even
in the case where the two variables have nothing to do with each other will have considerable
support on positive ranges of the interval zero therefore researchers may be misled they may be
misled by r squared to believe that they have found something important that they have found
some important relationship and finally peter philip showed that the germ watson statistic
converges to zero and this in the plim sense so when t goes to infinity then the plim of the
germ watson statistic is zero which is to say that the germ watson statistic correctly
indicates extremely high positive autocorrelation of the regression resilience you had
since the germ watson statistic converges against some fixed point namely zero we can pull it apart
by pre-multiplying it with something which does not converge in this course of course by something
which explodes which diverges in this case we multiply by t and not by square root of t
so if we look at the product between t the number of observations and the de
Watson then peter philip's show us that t times the de Watson again converges against distribution
converges in in distribution against some distribution which as you call d6
so the sixth type of non standard non degenerate distribution
which gives us the possibility of testing for a linear regression which is spurious
this is something which philip's also works out the hypothesis of a spurious regression
can be rejected for instance if the product of the number of observations t and the de
watson is greater than some critical value let's say five percent and this five percent
critical value must be found by looking at the non degenerate distribution d6
okay these are the results of peter philip's which are sort of the cornerstone of what we
know today about spurious regressions are there any questions relating to spurious regressions
we now come to the really important chapter on unit root tests because we need to test for
unit roots since we now know that if we work with time series without knowing whether they
are unit root processes then almost everything can happen which means leads us in believing
some kind of relationship is there whereas in fact there's no such thing and this is why i have
said two weeks ago that you can basically throw away all econometric studies which have been
made prior to peter philip's error so in the 1970s and early 1980s and before the 1970s of course
because we would need to check with the data people have used in these old papers
whether they have made used whether they have made use of unit root processes and if they did
then it is quite likely that their interpretation of their results was actually wrong
and that they didn't apply the correct tests so we know that in the spurious regression context
we can be easily misled and the spurious regressions problem comes up when we have
unit root in unit roots in our data not only in the random walk case but any type of unit root
process so we need to take great care to find out whether some macroeconomic time series
is in fact a unit root process which implies that we have to test for unit roots we have to
find some type of time series test which allows us to test whether time series does
or does not have unit root how do we do this well as i taught you already the two main models for
trending data are the model of trend stationarity and the model of different stationarity
where the trend is generated by drift in some type of unit root process
so the test which we construct for testing for unit root should be able to discriminate
between the two main models for trending data that we have had in our
the most famous of such trade tests is the so-called Dickey Fuller test
let us now discuss the Dickey Fuller test for the most primitive case first namely
the case of some ar1 process so we're familiar with that yt is equal to nu plus alpha yt minus
one plus ut in the first chapter of this lecture we have always assumed that alpha is smaller than
one in absolute value but precisely this is what we now would like to know is alpha possibly equal
to one so we have to think about the question how can we test the unit root hypothesis that
alpha is equal to one so the unit the null hypothesis would not be alpha is equal to zero
which is the standard null hypothesis which computer packages but the hypothesis is that
alpha is equal to one and we will later reformulate the equation in such a sense
that we can actually make use of the pre-programmed tests in econometric software
now when we want to test the unit root hypothesis alpha is equal to one
then we should make one important observation which is that there's no point in estimating
alpha equal to one in equation 18 if the data is trending so if we have trending data
then testing alpha equal to one in 18 does not make much sense because alpha equal to one is the
only possibility for this equation 18 here to admit trending data right there is no deterministic
trend in equation 18 therefore when we use equation 18 on trending data then the only
possibility for the data to display formally that they have a trend would be to set the alpha here
equal to one because then as we know the difference between yt and yt minus one is nu plus ut so
there's always some increase in the unit root series because the difference is on average
positive by the constant nu so nu is actually the parameter of the drift
we have reformulated equation 18 to see that actually do becomes the slow parameter of some
trend if we use this different stationary version of the equation so when we have trending data then
we cannot really test alpha equal to one because the data are forced to assume that alpha is equal
to one rather what we have to do and this is important to understand in the Dickie Fuller test
we have to specify an equation which can represent trending data both for the case that alpha is equal
to one and for the case that alpha is smaller than one absolutely which means that we have to
build a model which nests the two possibilities of trend stationarity and difference stationarity
now we do this in equation 19 on this slide suppose that the data are trending and then we
want to test the unit root hypothesis that alpha is equal to one in this model here well if alpha
is equal to one then or perhaps i should first explain this model here equation 19 is basically
equation 18 but allowing for a linear deterministic trend theta t as a new regressor so we have one
two regressors now plus the constant so if you also include the constant then there are three
regressors three regressors constant trend and lagged y if alpha is equal to one then the constant
will as we know already generate drift so in this case we do not need the theta t therefore
the null hypothesis would be that alpha is equal to one and that theta is equal to zero
we have to test both coefficients simultaneously actually right we have to see whether this is true
h1 is of course the contrary namely that alpha is smaller than one in absolute value
and that theta is different from zero okay so this is the general idea how we can test him
to estimate equation 19 now as an exercise show that alpha equal to one and theta
different from zero so show that if we do not have theta equal to zero in the case of alpha
equal to one but rather theta is different from zero that this would imply a quadratic trend
with the beta and then check your results by very fine that in the special case of theta equal to
zero you indeed get the new parameter as and parameter so there are various possibilities
to test for a unit and i give you four such possibilities here the possibility is that you
estimate equation 19 by ols and then you test the t statistic for the null hypothesis that
alpha is equal to one so you just exploit information on the estimated alpha to check
whether alpha is equal to one another possibility is that you estimate 19 by ols and then you test
the t statistic for the hypothesis that theta is equal to zero so for this part of the null hypothesis
so we test whether theta is equal to zero and if it were then this would imply that alpha
must be equal to one because otherwise we cannot accommodate training data
you can of course also combine this you can use an f statistic and simultaneously test in the
hypothesis that alpha is equal to one and theta is equal to zero or you can rewrite
the model a little bit and write the coefficient of the t which used to be theta as theta tilde
times one minus alpha everything else is unchanged in this regression well if we do it like this
then we have a non-linear estimation here because the alpha comes up as the coefficient of the one
and it comes up in the coefficient of the t here with the property that if alpha is equal to one
then obviously this whole thing one on this alpha is equal to zero times theta tilde is also equal
to zero so the coefficient of the t would be equal to zero the whole point of writing the equation
this way here is that if alpha is equal to one then automatically the coefficient on t is equal
to zero now since this is a linear relationship which we have here sorry non-linear relationship
in the parameters which we have here what we would need to do then is to use maximum likelihood
rather than a linear testing procedure like over s and then test the likelihood ratio statistic
for the hypothesis that alpha is equal to one and this would be the fourth possibility of a unit
fruit test in all four cases which i have just explained to you the t statistic is non-standard
so it is a statistic it's a distribution which i have never seen before and it was basically
derived like peter phillips derived since distributions here they were derived by dickey
to statisticians actually not economic traditions and not economists but statisticians who worked
on unit fruit processes and fuller was the professor and dickey was his PhD student he became
very famous by having contributed to the understanding of the spurious regressions
problem and for developing the test which allows us to test for unit fruits
because first dickey and fuller derived what kind of distribution the test statistics have
they showed it's not one of the standard distributions not normal distributions not
chi-square distribution t distribution f distribution so these are standard but rather
some other type of distribution nobody else nobody had previously worked with they gave
the formula how to compute this distribution and then they also tabulated the distribution by
indicating where are the five percent critical values um they were published by dickey and fuller
but their um yeah their their critical values are still a little imprecise computational
needs were just beginning to develop in in science and later it was mckinnon in 1991
who provided exacter critical values than dickey and fuller had done
but dickey full of the first do something like uh this um and mckinnon came much
later but still is useful to look at his book if you don't have the data already programmed
your econometric software which usually is in case today and dickey and fuller also tabulated
critical values for the f statistic but we will focus here on the t test in a in the previous
slide so on this t test here estimate by ols and test the t statistic for now hypothesis alpha is
equal to one and we will focus on this test because this is the most widely used in type
series econometrics now as you know by default econometric software computes a t statistic
and it does so for the hypothesis that the coefficient is zero so when you estimate
this model 19 here with usual econometric software then this software will give you
a t statistic for the alpha here and it will also give you a t statistic for the theta here
and the associated p values and you might be misled to think the computer is smart enough
to give you the correct p values but this is what the computer typically does not
do and by the way we are not interested in the hypothesis which the computer estimates
for us namely the hypothesis that alpha is equal to zero we want to test the hypothesis that alpha
is equal to one okay how can we do this well in order to make use of normal econometric
software in order to preserve what has been written and embodied in normal econometric
software we can rewrite our test equation a little bit so we subtract y t minus one
on both sides of equation 19 then we get delta y t which is of course
um does the difference between y t and y t minus one data y t is equal to nu plus theta t plus alpha
minus one times y t minus one plus ut so we just have subtracted y t minus one on the right hand
side and on the left hand side of the equation which then yields that we regress the growth
rate of y t on the leg level of y t
the hypothesis that alpha minus one is equal to zero on it this coefficient beta is equal to zero
is of course equivalent to the hypothesis that alpha is equal to one right if beta is equal to
zero then apparently alpha negative minus one is also equal to zero or we test alpha is equal to
this test is called the dicky fuller
so recall please the dicky fuller test involves that we regress the first difference on
of our dependent variable of y t on a constant on the linear trend and on the leg level y t minus one
so we do not um estimate a regression where we regress y t on y t minus one like we do in AR
processes but rather we regress a difference delta y t on the leg level y t minus one and
we have to have both the constant and the linear trend in the regression equation
now evaluate the t statistic for beta hat with the appropriate dicky fuller distribution that's the
next step in our test right we have to evaluate the t statistic for beta hat with the distributions
tabulated by dicky and fuller or with the better um simulation results by macular
we know that the null hypothesis that alpha is equal to one will be rejected when beta hat is
significantly different from zero and otherwise if it is not then we will just accept the null
here is another computer exercise which you should do write a computer problem which
runs let's say 10 000 times the following simulations generate 200 observations on a
random walk with drift drift and then estimate 20 by ols then compute the t statistic of beta
hat or beta take the t statistic which the regression software has given you for beta hat
and then collect the results for all those t statistics over the 10 000 iterations
now that you've done this plug your results in a histogram and check how often do you
observe um t statistics which are greater than two an absolute value and what is actually the
critical five percent value for a one-sided test yeah and then plot the t statistic the histogram
of the t statistics and of the estimated betas what you can also do is that you set the drift
term equal to say nu is equal to 0.2 and then um run the same type of test and this is what i
have done for you in this lecture so i ran again 100 000 runs of the simulations
in this exercise and show you here the results which i obtained
the t statistics for beta hat are distributed in this way here so this is the beta which is equal
to alpha minus one right so it is centered on zero apparently which is good news and seems
to be symmetric which is also good news sorry did i say no i'm sorry i think i did say centered on
zero this is not true it is not centered on zero right zero is here it is centered on some negative
number which is close to negative two right but zero is here which means that if we apply
conventional significance levels so if we say that um the coefficient is significant if the t
statistic is greater than two an absolute value then you see that more than half of our computed
t statistics here indicate a significance for this regression coefficient which is of course
much too much we would have here 50 or more likelihood for the t statistic to exceed two
rather than just five percent of such likelihood or actually for two it is 2.5 percent because
in normal inference we conduct a two-sided test whereas here as you see the second side of the test
namely the positive t statistic is rather unimportant we don't really want to know this
we would like to know how about negative
t statistic here and then we see that there are many results which indicate a t statistic which
is greater than two an absolute value and therefore seemingly significant
note that the estimated beta coefficient is distributed in a completely non-symmetrical way
right the estimated beta coefficient is usually negative very very rarely will you find a positive
beta one hand but mostly it is negative and you see it stretches out considerably
so in some sense it is clear that this thing here is absolutely not comparable to the kind
of distribution which we find in standard theory with stationary time series for the estimated
beta one hand the estimated constant is approximately centered on zero not quite
as you would see it is approximately symmetric but not quite as you see here already
at the center where the top of the distribution is not shaped in such a way that zero has
necessarily the greatest amount of likelihood if we now look at the distribution for the t
statistics of the constant term then you see a very funny and particular graph of this of this
distribution of the t statistics t statistics which are greater than two in many instances
and the significant value of the t statistic or the significance level of the t statistic
is actually in the range of 3.4 so somewhere here or somewhere well here approximately
the estimate of the trend coefficient should of course yield zero under the null hypothesis
but the estimate of the trend statistic as you see is zero in only very few cases and then again
it has a completely asymmetric distribution here it is clearly skewed to the right right
so it is quite likely that we obtain a positive estimate for the trend coefficient even in a case
where no no deterministic trend is in the data so another null hypothesis
if we look at the t statistic for the trend coefficient and try to determine the critical
value at the five percent level and we see that this critical value is approximately in the area
of 3.4 as well so just as the case of the estimated coefficient we find that here the critical value
is in the range of 3.4 so 3.4 is probably approximately here where my hand is and we
will find that then perhaps the rest of the probability mass is around five percent for
a one-sided test again not for a two-sided test as we do in standard inference
now let me give you an application to this for our series GDPT recall the per capita real GDP
in the United States that we have analyzed several times already and let's execute the
Dickey-Puller test for this time series GDPT so you know that the OLS estimation of this
regression equation of the test equation now proceeds in such such a way that we have to
regress the first difference of GDP so delta GDP on the lagged level so it's not the case
that the delta is missing here right or that this data is a misprint is completely correct
we have to estimate the first difference of GDPT on the lagged level of GDP and on a linear trend
and on a constant term so this is the test equation for the Dickey-Puller regression
and what we would find in this case is well the most important result is this result here
we estimate for beta hat a coefficient of negative 0.033 with a t statistic of negative 1.869
okay so this is a fairly high t statistic it's not quite two but we know that if t beta hat had a t
distribution the p value for such a t statistic of negative 1.869 under the null hypothesis for
the one-sided test would be 3.2 percent so in this case we would actually if we applied
usual inference standard inference we would say well this is too low and we reject the null
hypothesis because the p value is less than five percent but this is completely misleading because
the critical values of the one-sided Dickey-Fuller test on t beta hat are actually at the five percent
level negative 3.4 as I told you so I give you here the critical values at the one percent level
at the five percent level at the ten percent set level so that you can see what what happens
even at the ten percent level we have a t statistic which is greater than three in absolute numbers
so our t statistic of something even lower than two is far from being significant right so we must
conclude that the true value of the beta hat here is actually zero right and if the true value of
the beta hat is zero then recall beta is equal to alpha minus one so if beta is equal to zero
then alpha minus one is equal to zero this means that alpha is equal to one this means that we
have a real truth okay so it's the same thing basically happens here to the estimated coefficient
of the theta of the of the trend term so to the theta hat and you see the coefficient is quite
small but the t statistic with 1.8 is not that small at all but I showed you in my graph here
the t statistic for the trend coefficient has this type of distribution with a five percent critical
value being equal to approximately 3.4 but we have far of 3.4 here the t statistic is 1.8
so this is less than or this is little more than 50 percent of the 3.4 therefore we have to conclude
that this coefficient theta is not different from zero so in this case we again get the result which
we would find consistent with our result that beta hat is equal to zero because beta hat equal to
zero means that alpha is equal to one in the case that alpha is equal to one we do not need a
deterministic trend in the data if we have no use for them because as the exercise which I gave you
a little earlier implies that with alpha equal to one and non-zero theta we would have a quadratic
trend in the data and we know from the time series GDP that there is no quadratic trend
in the data so necessarily when the data decide for beta equal to zero or alpha equal to one
same thing right then the theta must be equal to zero and in fact the 1.8 here is something which
is insignificant so we can very easily accept the hypothesis that theta is equal to one as theta is
equal to zero when for the constant we do find something which is negative with a t statistic
which is negative 1.8 again in this case we would need a significant constant and this constant here
is not significant so this is somewhat difficult in terms of interpretation we would need a constant
which is different from we would need a constant which is significant because we have a trend in
the data and actually a positive trend so we actually need an estimate of the constant which
is positive but we don't get it here we get something which is negative and insignificant
so this seems to cause us some headache but the headache is not that big because currently we're
just testing and we're not so much interested in the magnitude of the constant here because we know
that our estimate of the constant is inefficient since we also allow for a linear trend so basically
what we have to do is that we first take these test results here and interpret them as indicating
very clearly that there is a unit root in the GDP series and since we know that there is a unit root
in the GDP series and actually the regression results here also suggest that the theta is equal
to zero we can then go ahead and estimate again regression where we estimate just the
delta GDP on a constant and on u2 right because we would know beta is equal to zero theta is equal
to zero so we can delete two regressors and thereby decrease the variance of our estimates
and therefore get much precisor estimates of the new here that we would have got here
okay yeah I think this is what I have to say here and perhaps we should break at this point
because before because now I come to the case of a random walk without a drift which has its own
intricacies and we should perhaps do this after the break so in five minutes time please
okay we come back are there any questions regarding the unit root testing procedure
so the Dickey Fuller test if so please raise your hand
okay and now just to make clear what I'm talking about the Dickey Fuller test which I just
described is a test for trending data or let's go perhaps back here to the test equation 19
yeah it was our thinking was that if we have a unit root with a drift then we necessarily have
to supply a trend regressor in order to give the data a chance to decide for a trend stationary
representation so for a representation where alpha would be smaller than one
and the trend is being explained by this trend term this is the
uh uh alternative hypothesis right given that the data are trending we need to supply
the data with a possibility to decide for the non-station for for the stationary
version so for the trend stationary version of this equation alpha smaller than one and theta
different from zero or of course under the null hypothesis they have the alternate they have the
possibility to decide for alpha equal to one and theta being equal to zero what usually is confusing
when students learn about these unit root tests at first is that we reformulate the test such
that we do not test the alpha equal to one but we test the beta being equal to zero but beta being
equal to zero is of course the same thing as alpha equal to one is if beta is defined as being alpha
minus one right so this involves as i have explained that we request delta yt on yt minus
one which is another sort of confusion but i'm sure you you can understand that well if you look
at these formula again at home it is absolutely necessary that you understand that well and
sometimes ask for these things in the accepts now um we did these simulations and uh now we want
to come to the other case where the data is not trending so the digi fuller test in the version
which i had just explained before the break was with trending data for trending data but obviously
there may also be cases of time series which do not have a trend and yet they may have a unit root
as you know so there may just be pure random walk without a drift so the case comes up how does the
unit or how can we test the unit root hypothesis or how do we have to shape the digi fuller case
in the case the digi fuller test in the case where uh the data do not have a trend
or the random walk does not have a drift
and this we want to study next and this is another sort of confusion or another form of
confusion which may unfortunately arise because there are it is important to distinguish these
cases with trend and without and with drift and without drift so in the case of a random walk
without drift we would have this type of process yt is equal to yt minus one plus ut right so we
know we can replace the yt minus one by yt minus two plus ut minus one plus ut and we can do this
many times so that eventually we get yt is equal to y zero plus the sum over
select u terms so plus cumulative 12 noise right
now the point is this is a random walk without a drift but it has a non-zero mean as we can see
unless the starting value y zero happens to be zero right but in the case where y zero is
different from zero we know that this is a unit root process which has a non-zero expected value
now you see that there was no constant in the initial equation here yt is equal to yt minus
one plus ut so there is no constant term in here but nevertheless we have to supply in our test
equation the constant even though there is no constant in this representation of the first line
because we know that yt may well have a non-zero mean right so we have to allow for a constant
under the alternative um yeah or perhaps i should tell you again um
if we test for alpha equal to one which is implicitly what i have assumed here right
there's one is the coefficient of the yt minus one here then the fact that alpha is equal to
one that there's one here is a coefficient already implies that the process may have a non-zero mean
however if the alpha is not equal to one if it is smaller than one then we know that the expected
value of yt would be zero unless we allow under the alternative for a constant
so the situation is actually analogous to the situation which we just had in the case of a
unit root process with a trend because if we have trending data then we know that we have to supply
in our test equation a trend term under the alternative but we do not need the trend term
and actually it would indicate raw trending behavior namely quadratic trending behavior
under the null hypothesis because under the null hypothesis the alpha being equal to one
already induces the drift in the data so we do not need a separate trend and here when we have
a random walk without a drift the same thing applies to the constant if alpha is equal to
one under the null hypothesis we do not need any constant to represent a non-zero mean of yt
because the non-zero mean would by virtue of these equations here already be embodied would
already be provided for but under the alternative that alpha is smaller than one we do need a
constant in order to accommodate for possibly non-zero mean of the data
so this implies that the test equation which is again the Dickey Fuller test equation it's just
another version of the Dickey Fuller test the test equation for non-trending data would be
delta yt is equal to nu plus beta times yt minus one plus ut so 21 equation 21 here is just a
special case of the test equation 19 which we had here right we have just deleted the theta t
otherwise it is actually the same equation yt is equal to constant plus alpha yt minus one
plus ut or if we rewrite it and subtract yt minus one on both sides of the equation we would have
delta yt here is equal to nu plus alpha minus one times yt minus one or if you again set beta
is equal to alpha minus one plus beta times yt minus one plus ut so that's the test equation
which i have just from here so this is equation 21 if we run this test then we will find that the
distribution of the beta here is different from the distribution of the beta in the case where
the test equation did allow for a deterministic linear trend so did allow for the term
beta times t so the critical values are different for this test here for non-trending data
then they are for trending data and the more general model where we do allow for deterministic
trend in the test equation so that's an important thing and as i say this is also sometimes
cause of confusion the critical values for the t statistic of beta dependent on whether or not
we include a trend in our test equation right in 21 we have not included trend term but in 19
or variant where we replaced the alpha by beta where we subtracted to yt minus one both sides
of the equation then we had allowed for a linear trend and therefore the critical values for the
beta term are different so try to convince yourself of the truth of what i've just said
by redoing the previous exercise for random work without the drift and then compare the
simulated five percent critical value for the one-sided test of the hypothesis that
beta is equal to zero so look at the distribution of the test statistics t beta
i give you an example for such a unit root test here using again the GDP data but now not the
level of the GDP data because the level of the GDP data is trending but the growth rate so the
first difference right i turn you the diagram of the plot i mean of data GDP t which is the growth
rate growth rates typically do not have a trend growth rates typically fluctuate around the
constant so since data GDP does not have a trend we cannot check whether data GDP has a unit root
whether the autoregressive representation of data GDP has a coefficient of alpha equal to one
or beta equal to zero we have now to apply the the fuller test to a regression equation
where we have on the left hand side the first difference of the dependent variable
since the different this is the dependent variable is delta GDP we have to difference
data GDP one more time to get data squared GDP t and this must be regressed on a constant
because as i explained to you under the alternative we do need a constant
and on data GDP t minus one which is now the level of the growth rate right so this is not
different one more time and beta of course is alpha minus one like before
i know of course we allow for some error term
so if i do this to my real burnt series GDP t then i estimate for beta hat negative 0.7
with the t statistic for this estimated coefficient of negative 11 basically 10.8
right the critical values for t beta hat in the one-sided test of the model for data without
trend are well given here for the five percent the level for instance negative 2.86
so the five percent level is still greater in absolute number then we would have it under a
normal t distribution uh where it would be uh 1.9 no not not 96 this is the two-sided level but
you you can dig it up what the exact level is but clearly smaller than this one yet here the
critical value is not as big an absolute size as it was um for the training model where we had
negative 3.4 negative 3.41 and so the critical value here is quite a bit smaller in absolute
value than in the training model and our conclusion is of course uh that with the t statistic of
negative 10.8 we are far in the in the significance uh significant reason region so the likelihood
under the null of such a t statistic is essentially zero right negative 10.8 is much much
smaller than negative 3.4 which would already indicate one percent significance so we conclude
the data GDP is clearly stationary the unit root hypothesis may be rejected for data GDP
please um make it clear to yourself that in unit root tests or at least in the digi fuller tests
we typically test the unit root uh we typically test the null hypothesis of a unit root so when
we cannot reject then we accept that there is a unit root in the data only if we can reject
the test then we assume uh that the data is stationary
okay any questions uh so far
now um if there are no questions then we come to this so-called augmented um digi fuller test
because the next question is what do we do if we do not have a simple random walk but some
more complicated unit root process so how do we test a higher order arp with p greater than 1
for unit roots suppose we have such a process here yt is equal to nu plus alpha 1 yt minus 1 plus
alpha 2 yt minus 2 plus plus plus alpha p yt minus p plus ut we know that such a process may
sorry may also have a unit root right um we know that depends on the sum of the alpha coefficients
when the sum of those alpha coefficients here is equal to 1 then we have a unit uh root in our
process the question is how can we test this well and this involves a little bit of uh rewriting
of equation 22 which may look a little complicated at first or is actually perhaps a little complicated
but the idea is fairly easy uh actually once you understood you'll see that is uh quite
quite easy so just follow me through those these calculations here what we can do is that we
rewrite this equation 22 such that we write mostly in first differences and we start with the last
term here alpha p yt minus p we want to make the last term some delta yt minus p plus 1 so
one period ahead in time what means ahead what one period not quite as uh far back in time as
yt minus p i try to have t minus p plus one and now we take the growth rate um here okay so the
first difference how can we achieve this well what i have done here is basically that i have just
written delta yt minus p plus one in place of yt minus p now i've changed the sign from plus
to a negative sign here observe that delta yt minus p plus one is equal to yt minus p plus one
minus yt minus p so the second term was minus yt minus p so minus this yt minus p here so that
the minus which is inside this delta being multiplied by the minus one here gives us
exactly again the alpha so plus alpha p yt minus p but what we then have done is that we have
apparently subtracted or newly introduced into this equation here a term negative alpha p times
yt minus p plus one right so this delta if you think of it as the difference between yt minus
p plus one and yt minus p and the yt minus p is what we had already in the equation here
but we have newly introduced negative alpha p yt minus p plus one so we have to add this
in the equation and we just do it in the preceding term the preceding term was alpha p minus one
times yt minus p plus one now we have to add an alpha p times yt minus p plus one to make
to to compensate for the alpha p yt minus p plus one which we have used here in order to
induce the first difference here okay so here is now the sum of alpha p minus one
and alpha p now in the next step we do the same thing again in the next step we use or we rewrite
this term here with a delta at this place right so delta yt minus p plus one and we change
the coefficient here so we make this alpha p minus one plus alpha p times delta yt minus p plus two
because there's a negative sign here and in the first difference here the yt minus p
plus one was also with a negative sign this accounts then for this term here and we have
to subtract again no to add again alpha p minus one plus alpha p times yt minus p plus two which
leads to in the previous term which is now here somewhere in the dots right in this term we would
get an alpha p minus two plus alpha p minus one plus alpha p okay and this way the whole
sum is contracted it'll construct it until eventually we arrive at this step here where
we introduce the delta in the yt minus term and then have a sum here over all the alphas
starting at i equal to two and going up to p well and then of course we can rewrite this again
and you should try to reproduce my calculations here at home by introducing a delta yt minus one
here and obtaining eventually the sum over all the alpha i times yt minus one here
now that's actually all we have to do if we look at this representation here
we see this is a representation of yt which has the sum of all the alpha coefficients here
ahead of or as a coefficient of the yt minus one and actually this is exactly the term we are
interested in because we want to test whether this sum here of all the alpha i's is equal to one
right and then we have all the other yt's in the regression is deltas with new coefficients
well you see here it's the sum over all the alphas from i going from two to p or here's
alpha p minus one plus alpha p well and here it's unchanged it's the alpha p
um now let's just call these new coefficients which have been built up here by new symbols
which are called alpha one bow and alpha p minus one bow so just to have shorthand notation for
that then we have this regression equation here yt is equal to a constant plus a coefficient
times yt minus one plus lots of coefficients which are coefficients of delta yt minus k's
right lots of them and we are interested in the t statistic for this thing here we are interested
to know if the sum of the coefficients here is different from one
this we do in the same way in which we have done it uh in the dikki poder test name we subtract
yt minus one on both sides so then we get delta yt is equal to nu plus the sum over all the alpha i's
minus one that we call again beta times yt minus one plus the sum of lagged differences of yt
right with certain coefficients these coefficients are typically not
important we don't really um look at them we don't really evaluate them we don't really test them
what we test is this beta coefficient again our hypothesis is that the sum of the alpha
coefficients is equal to one so one minus one would be equal to zero so we have to test again
or can test again whether beta is equal to zero so this equation 23 here is actually completely
analogous to the previous dikki poder test equation which was equation 20 uh with um still
theta being equal to zero right so i haven't covered the trending uh data case here in order
to keep the notation as simple as possible and it wasn't all that simple as you have seen so
this is why there was no trend in there but there is no particular difficulty of
cooperating uh in this type of of derivation patients so our coefficient beta which we will
test uh for what which we will want to check whether it is in fact zero um this coefficient
beta is the sum of all the alpha coefficients in the slack polynomial of our arp minus one
and so it is minus alpha of one where alpha of one is just the lag polynomial of
this process evaluated at point one so for l equal to one
yeah and then as i already said testing the null hypothesis beta the rely hypothesis that beta is
equal to zero is equivalent to testing the unit root hypothesis that alpha of one is equal to zero
so
essentially this equation 23 shows us that we construct a unit root test also for an arp
process just by regressing delta yt on yt minus one as we do in the standard dikki poder test
and supplying additional p minus one lagged differences as a regressor
this type of process or this type of test has again a different distribution for its t
statistics so the asymptotic distribution of the t statistic for beta hat is the dikki poder
distribution um so it is not a standard distribution but it is the same type of distribution as we have
added in the in the
Renowalk case but only asymptotically
in finite samples the distribution is different from the dikki poder distribution but it has
been derived and actually finite sample critical values have been simulated the reference here is
showing it lie they did this in 1995 so some 11 years after sine of dikki found the asymptotic
distribution of t beta had to be the dikki poder distribution and then showing it lie came along
and supplied critical values for finite samples and they are also typically embodied in software
like if you for instance
the regression equation must contain a constant
but this is not sufficient to cover for for trending data that's the same problem that we
had already in the um q renowalk case so if the data is trending uh then we also need a linear
trend and then the dikki poder the augmented dikki poder test equation would be delta yt
is a regression of the first difference of yt on a constant on a linear trend term to allow for a
non-stationary interpretation of the data on beta times yt minus one to test the null hypothesis
that actually the sum of the alpha i coefficients is equal to one plus then um the sum of beta i's
delta yt minus i's right so this is why the court test is called the augmented dikki
poder test because without this sum here we would have just the test equation of the dikki poder
test and now we augment this takes this um test equation by uh liked differences which are by
themselves not so very important uh we don't test these beta beta i coefficients here we can supply
as many of uh the delta yt minus i's as we find appropriate we should apply supply as many as uh
is required to make the regression residuals ut become uh quite so they should not be correlated
over time um the problem is of course that we don't really know what the true value of p is
but uh the adf test is also valid for unknown p um because side and dikki can show that
specifications with too many lag differences still yield consistent estimates under a number
of consistent conditions which i don't explain to you now because they are rather technical
but in principle we can just take any process which we suspect to be a unit root process which
we suspect to have a just um uh some of coefficients of uh one uh without knowing
how many coefficients we actually have to sum there and then apply the adf statistic and supply
a great number of lag differences and that would then lead to a very defined test
but of course incorrect specification of the p so if we have too many lag differences
increases then the variance of the estimates and therefore leads to a loss of power
in the adf test so um some um care is necessary uh some uh thorough working is necessary here
we have to make sure that we supply as many lag differences as the data requires to get
white noise residuals but on the other hand not too many of them because the power of
will then go down
what can we do to find the appropriate p well one possibility is of course that we
specify as many lag differences data y t minus i as we find uh significant in our regression
equation 24 um and the second possibility is that we specify as many lag differences
as is necessary to render the estimated regression residuals you had t uh white noise
so that there's no significant autocorrelation of ut hat anymore and the last possibility is the
use of information criteria i mean not the not the last in terms of real possibilities that's
quite a sensible way to proceed um but the last um here on my slides is to use information criteria
so either aka eq or schwarz criterion or hennen quinn or final prediction error
uh this doesn't really matter um but some information criteria to determine the optimal
likely so let us use um any in the following example so we just look at the t statistics of the
um data gdp terms so what i do here is that i test gdp t for unit roots
by estimating equation 24 with four lagged with four lagged differences and the j multi output
which i get here and which you would get similar in other econometrics uh softwares um is well the
gdp is what is denoted x in this output here so the lag level has a coefficient of negative 005
essentially with the t statistic of negative 2.7 and then there are the four uh lagged differences
so dx minus one dx minus two dx minus three and d of course positive and when we look at the t
statistics we see dx minus one is quite significant 3.72 dx is also significant um
dx minus two it's also also significant dx minus three seems to be uh unimportant um
the statistics very very low close to zero uh dx minus uh four is also insignificant right
so the lags one and two we certainly need but lags three and four is less clear
um if we delete the two lags in order to increase the precision of the estimates
then we get a new result here we see that the coefficient on x minus one so on the lag level
of the the regressor of the energies variable is now negative 0.048 with the t statistic of 2.78
the critical values are such that they correspond to the values in a model with trend of the
or the critical values are such we see that they are much bigger in size than and
then we have found it here so our conclusion would be that we cannot reject the null hypothesis
because because this t statistic here which would be significant at conventional levels
negative 2.8 is not big enough in absolute terms to be even 10 significant
now when we accept the null then we accept that beta is equal to zero but this of course
also implies that theta is equal to zero because otherwise as i already explained the data would
have a quadratic not the data the model would have a quadratic trend and typically we don't
see a quadratic trend in macroeconomic time series so after accepting the null with beta equal to zero
it is useful to re-estimate the model without a deterministic linear trend because also for the
linear trend of course it is clear that the linear trend if it is not necessary that then just
induces additional variance into our estimation and gives rise to less precise estimated coefficients
so we should now estimate the final model imposing a unit group and of course imposing a drift
this would mean that all the data are generated once are different once excuse me the lag level
disappears because there we have the sum over the alpha i's minus one but the sum of the other i's
is equal to one or to put differently the sum over the alpha i's minus one has been replaced by beta
and beta we found to be zero in our test we have a hypothesis that beta is equal to zero
but we don't need the lag level anymore and therefore we just have a regression in the
first differences so in the growth rates data gdpt is equal to nu plus beta one times data gdpt minus
one plus beta two data gdpt minus two plus ut and that's it no lag level no trend right
well we can of course uh estimate this what we would get then is a usual auto regression
where ggpt depends on gdp ggpt minus one with a t statistic of 3.54 so highly significant
and on ggpt minus two with a t statistic of 1.54 this is not so uh significant and therefore
it is not really clear whether we should retain this regressor in the regression equation so we
could actually make the next step and reduce the dimension of the regression equation even further
and then of course we have a significant constant here and what you do with the second
and late endogenous variable well this is a question for researcher
um yes now as i already uh uh said site and diki have studied um further extensions of uh the
tests developed by diki and fuller and what more result which they have obtained is the result that
the adf test is asymptotically valued valid even as a test for arma pq models uh why is this so
we have so far talked just over arp models why can we actually also use arma um pq data
generating writing processes or why can we fly it to arma pq data generating processes
well the intuition for this result is very easy actually we know an arma pq process if the ma
part is invertible can be inverted to yield an infinite order ar process and in this case we
would just need to approximate the infinite order ar process by a finite order ar process until no
further differences are significant or until the residuals are clearly white noise and then we can
justify the side diki extension of the diki fuller um tests just as we have applied them before
um a second remark on this is um the adf test removes autocorrelation in uh the
residuals parametrically parametrically means we supply lag differences to capture the autocorrelation
in uh the data um and uh these regressors the lag differences do this by means of certain
coefficients which we actually also give into the um regression equation um by uh certain
coefficients alpha i tilde alpha i bow which we have constructed so um there's a clear parameter
which accounts for a certain type of correlation over time of autocorrelation over time
there are two other authors um peter phillips again and his doctoral student um who in 1988
suggested a different unit group test based on non-parametric correction for residual autocorrelation
um these tests the phillips per all tests are often sometimes available in econometric
software for instance they are available in uh e views you can use them um but they depend on
some type of parameter some type of parameter k which you have to specify to um um specify how
strong the non-parametric correction uh should be done um and this gives some degree of freedom
and some degree of arbitrariness to the phillips for tests and this may be one reason why the
phillips per tests are less uh are often uh used but in principle they are valid tests for the
unit root hypothesis both type of tests reject the null hypothesis far too often in finite samples
um if the ma part is negatively correlated so if the ma part has negative autocorrelation
then um both types of tests reject the null far too often in finite samples is a simulation
studied by schwert in 1989 has uh indicated it is also important to know that not only the ADF
test but also the phillips peron test which i abbreviate here p p tests uh the null hypothesis
of a unit root so where we cannot reject then we accept the hypothesis of a unit root naturally
people have asked is it possible to also test the other way around because the result was that we
found many many unit root processes just because we were not able to reject the data and it may be
that some of these processes are not really unit root uh processes so the question came up can we
test the null hypothesis of stationarity against the alternative of um uh unit root well there were
four authors uh Kriatkowski again Peter Phillips Schmidt and Shin who in 1992 suggested a unit root
test based on the null of stationarity this test is then uh denoted the KPSS test for Kriatkowski,
Phillips Schmidt and Shin and Phillips is of course again Peter Phillips
both the phillips peron test and the KPSS test require the choice of a lag truncation parameter
and unfortunately there are no clear guidelines as to the correct price of this parameter
i mentioned this already as one of the drawbacks of the phillips peron test but the same drawback
exists for the KPSS test so what people usually want to do is that they run tests for various
choices of this parameter of this lag truncation parameter to see whether the results are robust
to the choice of the lag truncation parameter or whether perhaps uh the uh results depend strongly
on them and this of course would make things difficult.
The KPSS test is not implemented in all um time series econometrics uh software but you'll find
it in gmalti um you'll find it also in uh EU's and as i said the phillips peron test is also embodied in
EU's. As far as the lag truncation parameter is concerned EU's has also programmed a procedure
which automatically selects the lag truncation parameter this is then called bandwidth in uh
EU's which is perhaps not um the most unfortunate term to use because i don't think that all the
people would understand exactly what is meant by bandwidth.
If you have familiarized yourself with either gmalti or with uh EU's then please do the following
exercise and run the pp the phillips peron and the uh kpss EU test for the US GDP time series
of EU's which i gave to you uh at first use then the automatic bandwidth selection and then try
various user set choices to study the robustness and the aptitude of the automatic
bandwidth selection procedure which EU's currently uses.
Okay and here are some references which you may want to consult if further questions arise but
let me now there we when we are at the end of this uh set of slides first ask you if there are
any remaining questions concerning spurious regressions and EU tests please raise your
hand if there are any questions. Seems that this is not the case then let me move on to
slide three
where we now go back to stationary processes.
But uh now we want to cover not only univariate stationary processes but multivariate
variate stationary processes which actually is much more interesting from an economic
point of view because obviously in economics we usually talk about relationships between different
economic variables and economic theory gives us a number of such relationships which we may be
interested in testing with the data so as long as we just have one time series which we analyze
we're quite constrained actually in the type of questions that we can study. Now we want to move
on and allow that we study capital K variables simultaneously so we will now have a somewhat
different notation because yt will be a column vector consisting of k variables y1t up to ykt
so the prime here denotes the transposition right yt is a column vector y1t y2t y3t up to ykt
yeah and these are all the variables of interest that we may want to analyze particularly they can
be completely dependent or they may have lots of correlation with each other that actually is what
is interesting about this type of analysis that the variables do have something to do with that
we would like to find out what actually is it. So the instruments we now develop are called
vector autoregressions abbreviated VAR or just var and I start out here with the most simple
setting a var 1 so just get used to the vector notation here yt is now column vector y1t y2t
up to ykt for our capital K variables and similarly for ut that's also a column vector
for all the error terms in period t. I denote the constant by nu which is also a vector from nu1 nu2
to nu k and by a I denote all the coefficients which we now have so a is a capital K my capital
K matrix so it goes from a11 to a1k in the kth column of the matrix and it goes from a11 to a
k1 in the kth row of the series and please be aware that alpha 1k and alpha k1 are different
parameters or can be different parameters so the matrix a is not symmetric it's not like in
a covariance matrix. In my formula now I use zeros to indicate either a vector or a matrix
consisting completely of zeros right like it is usually in econometrics or matrix algebra
that we do not specifically write that something is zero matrix or a zero vector we just write a
zero and this is then assumed to be a zero of that form of the form which is compatible with
the dimensions of all the other vectors and matrices in our model.
Here are two definitions a k-dimensional process ut so this is this process here
you want to u2t ukt the k-dimensional process ut which satisfies that its expected value is zero
so is a vector of zeros and his covariance matrix is non-singular and the covariance matrix is u
times ut prime and I note again as sigma u so a process which has moments of this type first
mode is zero and second mode is some non-singular matrix sigma u with the property that there is no
correlation between ut and us is called a multivariate white noise process
yeah okay so important is here that we do not have auto correlation over time right between ut and us
now let ut be multivariate white noise we then have a k-dimensional process yt which satisfies
that yt is some constant vector plus a times yt minus one plus some error vector ut which actually
should be white noise and then this process is called a first order vector autoregressive
process or a bar one the nice thing about bars is that there are of course also higher order bars
but they can always be written as lower order bar namely as bar one the only thing which makes
which gives rise to a little bit of trouble then is that if we rewrite a higher order one
in the form of a bar one process then the covariance matrix of the higher order bar is
singular so the sigma u is then singular in order not to deal with this problem and go to the
bar one representation of our p data later we will currently confine our attention to the bar one case
so just do the easy stuff first so let's suppose as an example we have a
bar one process with two variables so capital k is equal to two then in standard notation an
explicit form without matrix notation we would have that our v consists of the two equations here
y1t depends on the constant nu one it depends on y1 t minus one also it's in past and it depends
on the past of the other variable y2 so it depends on y2 t minus one and y2 also depends on constant
which is now nu two and it depends on alpha two one y1 t minus one so on the past of the other
variable in the system and it depends on its own past and of course it depends on an error
this is slightly difficult because since the u1t and the ut are generally in general correlated
we would need a system estimator like a sure estimator sure you recall is a seemingly
unrelated regressions estimator unless we have both equations both the same type of regressors
and this is a result which was obtained already in the 1960s and as i write here by amald zener
who shows that when in such a system here where u1t and u2t are correlated we do not need those
two equations jointly if both regressions share exactly the same regressors and here we see that
the regressors are the same maybe y1 t minus one and y2 t minus one as in the second equation where
we also have y1 t minus one and y2 t minus one so only because the verb provides for the possibility
that all regressors appear as explanatory variables in all equations only this is why we
can estimate our equation by equation first estimate equation y1 and then estimate the equation
y2 t right so we do not need to go to some type of three-stage least square system estimator or
something like this which makes things rather complicated but rather we can just estimate the
whole valve equation by ols and one at a time so first the first equation then the second equation
then the last equation and so forth the third equation and so forth until we read the last
equation this seal equation estimate is then consistent because all the regressors are late
and they do not correlate with the error term right y1 t minus one and y2 t minus one cannot
correlate with u1 t because u1 t is white noise and it came into existence only in periods t
but not in period t minus one and two t minus one right so it can correlate with
the y1 and the y2 of the previous period and the same is true for the second equation here
this would of course be completely different if y1 t also depended on same period y2 t
or and vice versa right and unfortunately in economic systems this is usually the case
so i will give you in the next lecture an example of such a system which does not fit the bar
framework as i have introduced it now and then i will tell you what we can do about it and what
kind of cost comes along with it namely difficulties in interpreting the system
are there any remaining questions from your side
please raise your hand if this is the case
if it is not the case as it seems to be the case then say bye-bye until next week
you